Controllability Gramian - Revision history

92.201.151.66: /* See also */

2014-08-02T16:33:06Z

See also

← Older revision		Revision as of 18:33, 2 August 2014
Line 1:		Line 1:
	~~{{refimprove\|date=March 2009}}~~		Sportspersons Zelman from Ste. Anne, spends time with pursuits including legos, ganhando dinheiro na internet and riddles. Is a travel maniac and recently paid checking out the Rock Art of Alta.<br><br>Look at my web page: [http://ganhardinheiro.comoganhardinheiro101.com ganhe dinheiro]
	~~{{expert-subject\|Physics\|ex2=Energy\|ex3=Engineering\|date=March 2009}}~~
	~~In [[fluid dynamics]]~~, ~~'''turbulence kinetic energy''' ('''TKE''') is the mean [[kinetic energy]] per unit mass associated~~ with ~~[[Eddy (fluid dynamics)\|eddies]] in [[turbulent flow]]. Physically, the turbulence kinetic energy is characterised by measured [[root-mean-square]] (RMS) velocity fluctuations.~~

	In Reynolds-averaged Navier Stokes equations, the turbulence kinetic energy can be calculated based on the closure method, i.e. a [[Turbulence modeling\|turbulence model]]. Generally, the TKE can be quantified by the mean of the turbulence normal stresses:

	~~:<math> k = \frac12 \left( \overline{(u'_1)^2} + \overline{(u'_2)^2} + \overline{(u'_3)^2} \right). </math>~~

	TKE can be produced by fluid shear, friction or buoyancy, or through external forcing at low-frequency eddie scales(integral scale). Turbulence kinetic energy is then transferred down the turbulence energy cascade, and is dissipated by viscous forces at the [[Kolmogorov microscales\|Kolmogorov scale]]. This process of production, transport and dissipation can be expressed as:

	~~:<math> \frac{Dk}{Dt} + \nabla \cdot T' = P - \epsilon, </math>~~

	~~where:<ref>Pope, S. B., (2003) 'Turbulent Flows', Cambridge: Cambridge University Press</ref>~~

	* <math> Dk/Dt </math> is the mean-flow [[material derivative]] of TKE;
	* <math> \nabla \cdot T' </math> is the turbulence transport of TKE;
	* <math> P </math> is the production of TKE, and
	* <math> \epsilon </math> is the TKE dissipation.

	~~The full form of the TKE equation is~~
	~~:<math>~~
	~~\underbrace{ \frac{\partial k}{\partial t}}_{ \begin{smallmatrix}\text{Local}\\\text{derivative}\end{smallmatrix}}~~
	+
	~~\underbrace{\overline{u}_j \frac{\partial k}{\partial x_j}}_{ \begin{smallmatrix}\text{Advection}\end{smallmatrix}}~~
	~~= -~~
	~~\underbrace{ \frac{1}{\rho_o} \frac{\partial \overline{u'_i p'}}{\partial x_i} } _{ \begin{smallmatrix}\text{Pressure}\\\text{diffusion}\end{smallmatrix}}~~
	-
	~~\underbrace{ \frac{\partial \overline{k u_i'}}{\partial x_i} }_{ \begin{smallmatrix} \text{Turbulent}\\ \text{transport} \\ \mathcal{T} \end{smallmatrix}}~~
	~~+ \underbrace{ \nu\frac{\partial^2 k}{\partial x^2_j} }_{\begin{smallmatrix} \text{Molecular}\\ \text{viscous}\\ \text{transport} \end{smallmatrix}}~~
	~~\underbrace{ - \overline{u'_i u'_j}\frac{\partial \overline{u_i}}{\partial x_j} }_{\begin{smallmatrix} \text{Production}\\ \mathcal{P} \end{smallmatrix}}~~
	- \underbrace{ \nu \overline{\frac{\partial u'_i}{\partial x_j}\frac{\partial u'_i}{\partial x_j}} }_{\begin{smallmatrix} \text{Dissipation}\\ \epsilon_k \end{smallmatrix}}
	~~- \underbrace{ \frac{g}{\rho_o} \overline{\rho' u'_i}\delta_{i3} }_{\begin{smallmatrix} \text{Buoyancy flux}\\ b \end{smallmatrix}}~~
	~~</math>~~

	~~By examining these phenomena, the turbulence kinetic energy budget for~~ a particular flow can be found.<ref>Baldocchi, D. (2005), ''[http://nature.berkeley.edu/biometlab/espm129/notes/Lecture%2016%20Wind%20and%20Turbulence%20Part%201%20Surface%20Boundary%20Layer%20Theory%20and%20Principles%20notes.pdf Lecture 16, Wind and Turbulence, Part 1, Surface Boundary Layer: Theory and Principles ]'', Ecosystem Science Division, Department of Environmental Science, Policy and Management, University of California, Berkeley, CA: USA.</ref>

	~~==Computational fluid dynamics==~~

	In [[computational fluid dynamics]] (CFD), it is impossible to numerically simulate turbulence without discretising the flow-field as far as the [[Kolmogorov microscales]], which is called [[direct numerical simulation]] (DNS). Because DNS simulations are exorbitantly expensive due to memory, computational and ~~storage overheads, turbulence models are used to simulate~~ the ~~effects of turbulence. A variety~~ of ~~models are used, but generally TKE is a fundamental flow property which must be calculated in order for fluid turbulence to be modelled~~.

	~~===Reynolds-averaged Navier–Stokes equations===~~
	~~[[Reynolds-averaged Navier–Stokes equations\|Reynolds-averaged Navier–Stokes]] (RANS) simulations use the Boussinesq [[eddy viscosity]] hypothesis~~ <~~ref~~>~~Boussinesq, J. (1877) ‘Théorie de l’Écoulement Tourbillant.’ Mem. Présentés par Divers Savants Acad. Sci. Inst. Fr. 23, 46-50~~<~~/ref> to calculate the Reynolds stresses that result from the averaging procedure:~~

	~~:<math~~> ~~\overline{u'_i u'_j} = 2/3 k \delta_{ij} - \nu_t \left( \frac{\partial \overline{u_i}}{\partial x_j} + \frac{\partial \overline{u_j}}{\partial x_i} \right), </math>~~

	~~where~~

	~~:<math> \nu_t = c \cdot k^{1/2} l_m. </math>~~

	~~The exact method of resolving TKE depends upon the turbulence model used; ''[[k-ε turbulence model\|k-ε]]'' (k–epsilon) models assume isotropy of turbulence whereby the normal stresses are equal:~~

	~~:<math> \overline{u'^2} = \overline{v'^2} = \overline{w'^2}. </math>~~

	This assumption makes modelling of turbulence quantities (''k'' and <math>\epsilon</math>) simpler, but will not be accurate in scenarios where anisotropic behaviour of turbulence stresses dominates, and the implications of this in the production of turbulence also leads to over-prediction since the production depends on the mean rate of strain, and not the difference between the normal stresses (as they are, by assumption, equal).<ref>Laurence, D. (2002) ‘Applications of Reynolds Averaged Navier Stokes Equations to Industrial Flows.’ In van Beeck, J. P. A. J., Benocci, C. (eds), Introduction to Turbulence Modelling, Held March 18–22, 2002 at ~~Von Karman Institute for Fluid Dynamics. Rhode St.Genese, Belgium~~: ~~Von Karman Institute for Fluid Dynamics</ref>~~

	[[Reynolds stresses\|Reynolds-stress]] models (RSM) use a different method to close the Reynolds stresses, whereby the normal stresses are not assumed isotropic, so the issue with TKE production is avoided.

	~~===Boundary conditions===~~
	Accurate prescription of TKE as boundary conditions in CFD simulations are important to accurately predict flows, especially in high Reynolds-number simulations. Some possible examples are given below.

	~~:<math> k = \frac32 ( U I )^2, </math>~~

	~~where <math> I </math> is the initial turbulence intensity [%] given below, and~~

	~~:<math> U </math> is the initial velocity magnitude;~~
	~~:<math> \epsilon = {c_\mu}^{3/4} k^{3/2} l^{-1}. </math>~~

	~~Here <math>l</math> is the turbulence or eddy length scale, give below, and~~

	~~:<math> {c_\mu} </math> is a ''k-ε'' model parameter whose value is typically given as 0.09;~~
	~~:<math> I = 0.16 Re^{-1/8}. </math>~~

	~~Further <math>Re</math> is the Reynolds number~~

	~~:<math> l = 0.07L, </math>~~

	with <math>L</math> a characteristic length. For internal flows this may take the value of the inlet duct (or pipe) width (or diameter) or the hydraulic diameter <ref>Flórez-Orrego et al (2012) ‘Experimental and CFD study of a single phase cone-shaped helical coiled heat exchanger:
	~~an empirical correlation’ PROCEEDINGS OF ECOS 2012 - THE 25 TH INTERNATIONAL CONFERENCE ON~~
	~~EFFICIENCY, COST, OPTIMIZATION, SIMULATION AND ENVIRONMENTAL IMPACT OF ENERGY SYSTEMS, JUNE 26-29, 2012, PERUGIA, ITALY (ISBN online: 978-88-6655-322-9)</ref>~~ [http://~~books~~.~~google~~.com.br/books?id=gumvHDQmJD0C&pg=PA302&lpg=PA302&dq=Experimental+and+CFD+study+of+a+single+phase+cone-shaped+helical+coiled+heat+exchanger%3A+an+empirical+correlation&source=bl&ots=c7N1itAiOI&sig=-nydeNk9hXSTt8CMYSMB0hsVRlg&hl=en&sa=X&ei=iTF_UfKXE6vH0AHRiYGgCg&redir_esc=y#v=onepage&q=Experimental%20and%20CFD%20study%20of%20a%20single%20phase%20cone-shaped%20helical%20coiled%20heat%20exchanger%3A%20an%20empirical%20correlation&f=false].

	~~==References==~~
	~~{{reflist}}~~


	~~==External links==~~
	* [http://www.cfd-online.com/Wiki/Introduction_to_turbulence/Turbulence_kinetic_energy Turbulence kinetic energy] at CFD Online.
	*Absi R., Analytical solutions for the modeled k-equation, ASME J. Appl. Mech. 75 (2008) 044501,1-4. DOI: 10.1115/1.2912722
	~~[[Category:Computational fluid dynamics]]~~
	~~[[Category:Turbulence]]~~
	~~[[Category:Energy (physics)]~~]

en>Applebytom: /* External links */ Adding link to Mathematica's function page

2013-10-16T21:59:53Z

External links: Adding link to Mathematica's function page

← Older revision		Revision as of 23:59, 16 October 2013
Line 1:		Line 1:
	~~I would like to introduce myself to you~~, ~~I am Andrew and my wife doesn~~'~~t like it~~ at ~~all~~. ~~Alaska~~ is ~~exactly~~ where he'~~s always been living~~. ~~To perform lacross~~ is ~~some thing I truly enjoy performing~~. ~~Since he was eighteen he~~'~~s been working~~ as ~~an information officer~~ but ~~he ideas~~ on ~~changing it~~.<br><br>~~Feel free~~ to ~~surf~~ to ~~my web page~~ :: ~~are psychics real~~, [~~https~~://~~www~~.~~machlitim~~.~~org~~.il/~~subdomain~~/~~megila~~/~~end~~/~~node~~/~~12300 similar internet page~~],		{{refimprove\|date=March 2009}}
			{{expert-subject\|Physics\|ex2=Energy\|ex3=Engineering\|date=March 2009}}
			In [[fluid dynamics]], '''turbulence kinetic energy''' ('''TKE''') is the mean [[kinetic energy]] per unit mass associated with [[Eddy (fluid dynamics)\|eddies]] in [[turbulent flow]]. Physically, the turbulence kinetic energy is characterised by measured [[root-mean-square]] (RMS) velocity fluctuations.

			In Reynolds-averaged Navier Stokes equations, the turbulence kinetic energy can be calculated based on the closure method, i.e. a [[Turbulence modeling\|turbulence model]]. Generally, the TKE can be quantified by the mean of the turbulence normal stresses:

			:<math> k = \frac12 \left( \overline{(u'_1)^2} + \overline{(u'_2)^2} + \overline{(u'_3)^2} \right). </math>

			TKE can be produced by fluid shear, friction or buoyancy, or through external forcing at low-frequency eddie scales(integral scale). Turbulence kinetic energy is then transferred down the turbulence energy cascade, and is dissipated by viscous forces at the [[Kolmogorov microscales\|Kolmogorov scale]]. This process of production, transport and dissipation can be expressed as:

			:<math> \frac{Dk}{Dt} + \nabla \cdot T' = P - \epsilon, </math>

			where:<ref>Pope, S. B., (2003) 'Turbulent Flows', Cambridge: Cambridge University Press</ref>

			* <math> Dk/Dt </math> is the mean-flow [[material derivative]] of TKE;
			* <math> \nabla \cdot T' </math> is the turbulence transport of TKE;
			* <math> P </math> is the production of TKE, and
			* <math> \epsilon </math> is the TKE dissipation.

			The full form of the TKE equation is
			:<math>
			\underbrace{ \frac{\partial k}{\partial t}}_{ \begin{smallmatrix}\text{Local}\\\text{derivative}\end{smallmatrix}}
			+
			\underbrace{\overline{u}_j \frac{\partial k}{\partial x_j}}_{ \begin{smallmatrix}\text{Advection}\end{smallmatrix}}
			= -
			\underbrace{ \frac{1}{\rho_o} \frac{\partial \overline{u'_i p'}}{\partial x_i} } _{ \begin{smallmatrix}\text{Pressure}\\\text{diffusion}\end{smallmatrix}}
			-
			\underbrace{ \frac{\partial \overline{k u_i'}}{\partial x_i} }_{ \begin{smallmatrix} \text{Turbulent}\\ \text{transport} \\ \mathcal{T} \end{smallmatrix}}
			+ \underbrace{ \nu\frac{\partial^2 k}{\partial x^2_j} }_{\begin{smallmatrix} \text{Molecular}\\ \text{viscous}\\ \text{transport} \end{smallmatrix}}
			\underbrace{ - \overline{u'_i u'_j}\frac{\partial \overline{u_i}}{\partial x_j} }_{\begin{smallmatrix} \text{Production}\\ \mathcal{P} \end{smallmatrix}}
			- \underbrace{ \nu \overline{\frac{\partial u'_i}{\partial x_j}\frac{\partial u'_i}{\partial x_j}} }_{\begin{smallmatrix} \text{Dissipation}\\ \epsilon_k \end{smallmatrix}}
			- \underbrace{ \frac{g}{\rho_o} \overline{\rho' u'_i}\delta_{i3} }_{\begin{smallmatrix} \text{Buoyancy flux}\\ b \end{smallmatrix}}
			</math>

			By examining these phenomena, the turbulence kinetic energy budget for a particular flow can be found.<ref>Baldocchi, D. (2005), ''[http://nature.berkeley.edu/biometlab/espm129/notes/Lecture%2016%20Wind%20and%20Turbulence%20Part%201%20Surface%20Boundary%20Layer%20Theory%20and%20Principles%20notes.pdf Lecture 16, Wind and Turbulence, Part 1, Surface Boundary Layer: Theory and Principles ]'', Ecosystem Science Division, Department of Environmental Science, Policy and Management, University of California, Berkeley, CA: USA.</ref>

			==Computational fluid dynamics==

			In [[computational fluid dynamics]] (CFD), it is impossible to numerically simulate turbulence without discretising the flow-field as far as the [[Kolmogorov microscales]], which is called [[direct numerical simulation]] (DNS). Because DNS simulations are exorbitantly expensive due to memory, computational and storage overheads, turbulence models are used to simulate the effects of turbulence. A variety of models are used, but generally TKE is a fundamental flow property which must be calculated in order for fluid turbulence to be modelled.

			===Reynolds-averaged Navier–Stokes equations===
			[[Reynolds-averaged Navier–Stokes equations\|Reynolds-averaged Navier–Stokes]] (RANS) simulations use the Boussinesq [[eddy viscosity]] hypothesis <ref>Boussinesq, J. (1877) ‘Théorie de l’Écoulement Tourbillant.’ Mem. Présentés par Divers Savants Acad. Sci. Inst. Fr. 23, 46-50</ref> to calculate the Reynolds stresses that result from the averaging procedure:

			:<math> \overline{u'_i u'_j} = 2/3 k \delta_{ij} - \nu_t \left( \frac{\partial \overline{u_i}}{\partial x_j} + \frac{\partial \overline{u_j}}{\partial x_i} \right), </math>

			where

			:<math> \nu_t = c \cdot k^{1/2} l_m. </math>

			The exact method of resolving TKE depends upon the turbulence model used; ''[[k-ε turbulence model\|k-ε]]'' (k–epsilon) models assume isotropy of turbulence whereby the normal stresses are equal:

			:<math> \overline{u'^2} = \overline{v'^2} = \overline{w'^2}. </math>

			This assumption makes modelling of turbulence quantities (''k'' and <math>\epsilon</math>) simpler, but will not be accurate in scenarios where anisotropic behaviour of turbulence stresses dominates, and the implications of this in the production of turbulence also leads to over-prediction since the production depends on the mean rate of strain, and not the difference between the normal stresses (as they are, by assumption, equal).<ref>Laurence, D. (2002) ‘Applications of Reynolds Averaged Navier Stokes Equations to Industrial Flows.’ In van Beeck, J. P. A. J., Benocci, C. (eds), Introduction to Turbulence Modelling, Held March 18–22, 2002 at Von Karman Institute for Fluid Dynamics. Rhode St.Genese, Belgium: Von Karman Institute for Fluid Dynamics</ref>

			[[Reynolds stresses\|Reynolds-stress]] models (RSM) use a different method to close the Reynolds stresses, whereby the normal stresses are not assumed isotropic, so the issue with TKE production is avoided.

			===Boundary conditions===
			Accurate prescription of TKE as boundary conditions in CFD simulations are important to accurately predict flows, especially in high Reynolds-number simulations. Some possible examples are given below.

			:<math> k = \frac32 ( U I )^2, </math>

			where <math> I </math> is the initial turbulence intensity [%] given below, and

			:<math> U </math> is the initial velocity magnitude;
			:<math> \epsilon = {c_\mu}^{3/4} k^{3/2} l^{-1}. </math>

			Here <math>l</math> is the turbulence or eddy length scale, give below, and

			:<math> {c_\mu} </math> is a ''k-ε'' model parameter whose value is typically given as 0.09;
			:<math> I = 0.16 Re^{-1/8}. </math>

			Further <math>Re</math> is the Reynolds number

			:<math> l = 0.07L, </math>

			with <math>L</math> a characteristic length. For internal flows this may take the value of the inlet duct (or pipe) width (or diameter) or the hydraulic diameter <ref>Flórez-Orrego et al (2012) ‘Experimental and CFD study of a single phase cone-shaped helical coiled heat exchanger:
			an empirical correlation’ PROCEEDINGS OF ECOS 2012 - THE 25 TH INTERNATIONAL CONFERENCE ON
			EFFICIENCY, COST, OPTIMIZATION, SIMULATION AND ENVIRONMENTAL IMPACT OF ENERGY SYSTEMS, JUNE 26-29, 2012, PERUGIA, ITALY (ISBN online: 978-88-6655-322-9)</ref> [http://books.google.com.br/books?id=gumvHDQmJD0C&pg=PA302&lpg=PA302&dq=Experimental+and+CFD+study+of+a+single+phase+cone-shaped+helical+coiled+heat+exchanger%3A+an+empirical+correlation&source=bl&ots=c7N1itAiOI&sig=-nydeNk9hXSTt8CMYSMB0hsVRlg&hl=en&sa=X&ei=iTF_UfKXE6vH0AHRiYGgCg&redir_esc=y#v=onepage&q=Experimental%20and%20CFD%20study%20of%20a%20single%20phase%20cone-shaped%20helical%20coiled%20heat%20exchanger%3A%20an%20empirical%20correlation&f=false].

			==References==
			{{reflist}}


			==External links==
			* [http://www.cfd-online.com/Wiki/Introduction_to_turbulence/Turbulence_kinetic_energy Turbulence kinetic energy] at CFD Online.
			*Absi R., Analytical solutions for the modeled k-equation, ASME J. Appl. Mech. 75 (2008) 044501,1-4. DOI: 10.1115/1.2912722
			[[Category:Computational fluid dynamics]]
			[[Category:Turbulence]]
			[[Category:Energy (physics)]]

en>David Eppstein: /* External links */ stub sort

2012-08-29T07:21:55Z

External links: stub sort

New page

I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Alaska is exactly where he's always been living. To perform lacross is some thing I truly enjoy performing. Since he was eighteen he's been working as an information officer but he ideas on changing it.<br><br>Feel free to surf to my web page :: are psychics real, [https://www.machlitim.org.il/subdomain/megila/end/node/12300 similar internet page],